Orbitz And Keplers Laws Tutorial Mastering Astronomy Assignment

15) What conditions are required for a solar eclipse (hint: phase of the moon and nodes of the Moon’s orbit)? The phase of the moon must be new, and the nodes of the moon's orbit must be nearly aligned with the earth and the sun 16) What conditions are required for a lunar eclipse (hint: phase of the moon and nodes of the Moon’s orbit)? The phase of the Moon must be full, and the Moon's orbital plane must lie in the ecliptic.17) What causes the apparent retrograde motion of the planets? As Earth passes another planet, the other planet appears to move backward with respect to the background stars, but the planet's motion does not really change. 18) If the Earth's rotation axis was tilted by 45 degrees instead of 23.5 degrees, how would that affect the climate of the Earth? Would the seasons be more or less extreme? Why? more extreme summers and winters19) What do astronomers mean by a constellation? A constellation is a region in the sky as seen from Earth. 20) What is the celestial sphere? The celestial sphere is a representation of how the entire sky looks as seen from Earth. Chapter 3: the science of astronomy 21) What does Kepler’s third law say? If an asteroid has an orbital period of 8 years,what is the average distance of the asteroid from the Sun in astronomical units? Planets with larger average orbital distances have longer orbital periods. The ratio of the squares

Presentation on theme: "Announcements HOMEWORK assignment 2 due Feb 16 See masteringastronomy.com Assignment contains two tutorials; eclipses, phases of the moon Things like."— Presentation transcript:


2 Announcements HOMEWORK assignment 2 due Feb 16 See masteringastronomy.com Assignment contains two tutorials; eclipses, phases of the moon Things like unopened hint bonus, allowable attempts per answer all changes as the semester progresses, these are written on the first page of the homework assignment First Mid-Term Exam will be Feb 19 (Mon) Second Mid-Term Exam will be Mar 14 (Weds)

3 Part 1: Early Cosmological Ideas Topics : Introduction The Birth of "Scientific" Cosmology Aristotle & Ptolemy The "State of the Universe" for the Greeks The "Dark" Millennium The Recovery of W. European Science

4 Introduction The word Cosmology is from the Greek kosmos (world) and logia (from legein: to speak). All civilizations (probably) developed some form of "cosmology". Indeed all civilizations (we know about) seem to have "Creation myths" of some sorts. In the earliest civilizations (& up to very recent times) cosmology was primarily (often totally) a branch of religion/myth.

5 Introduction Certainly many (all ?) ancient civilizations performed astronomical observations of various levels of sophistication - from the neolithic observatories (eg.Stonehenge), to written reports (eg. The Chinese, Egyptians, etc).

6 ..a little humility required ? AND we should not underestimate the struggles/difficulties early thinkers had attempting to explain the universe within their various cultural, religious & technological environments …. Many earlier beliefs & models may seem "silly" or "absurd" now. Our beliefs and our scientific methods are "clearly far superior" - aren't they ? However it worth reminding ourselves from the very start that we have not figured everything out yet !! (dark matter, cosmic accel n, WIMPs, MACHOS etc) Many variations on Mythological/Cosmological Ideas, but many with themes not so different from our own:

7 ..a little humility required ? Imagine life before telescopes… Earth seems flat and motionless Sun, Moon, planets, stars move in sky (East to West) Strange things appear (comets, meteors) The Greeks are generally credited with promoting the idea that the universe was understandable using logic, and could be described by mathematics. This is a huge leap forward, and the basis of all science today

8 Birth of "Scientific" Cosmology By ~400BCE, generally thought that the universe worked & evolved through "natural" processes that can be observed on Earth....Divine intervention is not required (at least in the "running" of the universe). Cosmology was really a branch of philosophy at the time, but the Empirical Scientific Method was developing. Consensus - there are no limits to what can be observed & understood (again, at least not concerning the "running" of the universe), theories could be postulated, predictions made, theories revised as necessary

9 Birth of "Scientific" Cosmology The Greeks understood that the Earth is a sphere:- Observation of ships sailing over the horizon Observation of Earths shadow on the moon during lunar eclipses

10 Geocentric Spheres Only five planets were known to the Early Greeks (Mercury, Venus, Mars, Saturn, Jupiter). Thus early Greek cosmologists believed they had to account for 8 celestial entities -the 5 planets, the Sun, the Moon, and the "Stars". Cosmologies were all "naturally" Geocentric (centered on the Earth) Cosmologies generally included perfect spheres ( Sphere count: Pythagoras of Samos,(c.530BCE) 8)

11 Aristotle (c.350 BCE) The physical universe was finite - beyond the outer sphere of the stars was the (non-material) spiritual realm. Reasoned that that this "universe" must be unique (& have a single center), and have existed for eternity (& in a steady-state). ( Sphere count: Aristotle,(c.350BCE) 55) Believed celestial bodies move in perfect circles Forced to increase the number of spheres due to refined observations of planetary motion

12 Hipparchus (c.125 BCE) Refined distances to (& hence size of) the Moon (via Parallax )...made the first step determining the scale of the "cosmos" Suggested that the Sun appeared to be much larger than the Earth....some aesthetic concerns for a geocentric universe … but these were generally ignored… Constructed a catalog of close to 1000 stars. Discovered precession (1 degree/century) - the change of the position of the stars with time (now known to be due to the precession of the Earth's axis). In 134 BCE he discovered a new star (a nova),...in direct contradiction to the paradigm that the "heavens" were unchanging.

13 Parallax Larger parallax = smaller distance...the apparent change of position of a (closer) object as measured against the positions of more distant object(s) due to the movement of the observer. A Parsec is defined as the distance of an object that exhibits parallax of 1 arcsec (Easy to remember since the word parsec is a construction from parallax and arcsec) 1 parsec = 3.085678 x 10 16 m = 3.26 light years

14 Hipparchus, Parallax & the Moon Since 20% of the Sun's disk corresponds to 6 arcmins, then by estimating the distance between the cities one can derive the distance of the Moon. Hipparchus of Rhodes estimated the distance to the Moon from measurements taken during a solar eclipse in189BCE. eclipse was "full" in Hellespont (NW. Turkey), partial in Alexandria (Egypt) 20% of the Sun's disk remained visible in Alexandria Hipparchus estimated distance (4.5 to 5.2) x 10 8 m (c.f. modern value of 3.8 x 10 8 m) First attempt to scale the cosmos

15 Ptolemy (c.150) His (13 volume) master work Megale Syntaxis ("Great Compilation") Is usually known as by the arab translation Almagest ("The Greatest") Extended the system of Aristotle, sticking "religiously" to the ideas of a geocentric cosmology, the perfection of spheres a finite universe. (There is an on-going debate whether he stole/plagiarized Hipparachus' data !) [e.g. see Schaefer, Sky & Telescope 2002 Feb issue, p39]

16 Ptolemy & Epicycles The key elements are the epicycle to account for the retrograde motion of the planets the deferent (main circle) to account for the brightening & speeding-up of the planets at some times. relative tilts between the various planes, Large number of parameters (for 8 celestial objects) The “equant” is the point from which ang. vel of epicycle ~const) (Bothun, Fig1.2) The model was able to make accurate predictions Remained the "standard" cosmological model for 1400 years

17 Math as a Parameterization Contrary to the Greek ideas that Nature is simple, perfect, beautiful In the middle ages some thinkers started suggesting that the Ptolemaic system of eccentrics/epicycles do not actually exist,...but are merely convenient mathematical descriptions of celestial motion & reality (not reality itself) Note: The concept of Nature being simple, perfect, beautiful can be argued to be back with some modern theories (!)

18 The Ptolemy Monopoly Alan W. Hirshfeld, in "Parallax - The Race to Measure the Cosmos" “In a sense, Ptolemy was the Bill Gates of his day. His Ptolemaic "operating system", despite its known deficiencies, grew to dominate - in fact, monopolize - the astronomical market place."

19 Rise of the Scientific Method The modern Scientific Method observations, theory, predictions, tests/revisions... William of Ockham (or Occam; c.1300) suggests entities should not be “multiplied unneccessarily”, leading to (interpreted as) Ockham's Razor: the simplest (most succinct) theory is more likely to be correct, and certainly a better working model (to attempt) to disprove first. Roger Bacon (c.1250) helps popularize the scientific method in W.Europe,

20 Other Early Cosmologies (?) It should be remembered that our knowledge of history is solely dependent us on having written records Also few (if any) of the original works survive, so we must rely on later works (true & complete reporting ?) Who knows what ideas have been lost... One (radical) idea that was not developed (apparently ignored) is due to Aristarchus (c280BCE - between Aristotle & Ptolemy) a Heliocentric universe - the Earth orbiting the Sun (!)

21 Recap The following should be remembered: Cosmology one of the oldest philosophies/sciences Many ancient cosmologies grappled with some of the same deep philosophical questions we still ponder with today. The Greeks first (we think) reasoned that Universe was formed by natural processes which could be observed, understood/explained by mathematics Developed the Empirical Scientific Method Developed a geocentric system (Pythagoras of Samos, c.550BCE; Aristotle,c.350BCE) culminating with that of Ptolemy,c.150 involving a complex arrangement of spheres & epicycles. Reason and beauty/perfection were a strong influence of their thoughts. The universe was reasoned to be finite but eternal/unchanging

22 Recap Greeks thought Earth was stationary, if it were moving, wouldn’t we feel a sense of motion (great winds, loose objects whizzing by us etc)

23 Recap (cont) Mathematics (ie. the Ptolemaic system) seen as a parameterization, By c.1400, the Ptolemaic (geocentric) system had remained essentially unchallenged as the cosmology for 1300 years You should be familiar with the concept of Parallax the basics of how the Ptolemaic system works (how epicycles, deferent etc account for retrograde motion). the concept of Ockham's Razor Again, a detailed knowledge of names, dates and places is not required However, you should be familiar with at least the names & approximate dates of Aristotle (c.350BCE) and Ptolemy (c.150).

24 Foundat n of Modern Cosmo Continuing… The Earth moves from Center Stage And Then the Apple Dropped… Summary at the beginning of the C20 th

25 State of the Universe, 1400 By 1400, the geocentric cosmology of Aristotle & Ptolemy (based on concentric spheres, epicycles etc) had been essentially unchallenged for well over a thousand years. However, in the 15th & 16th centuries, following the years of the "Black Death" & centuries of strife, the start of the Renaissance in W.Europe finally allowed scientific & technological progress.

26 Rumblings of Discontent In c.1430, Nicholas de Cusa published On Learned Ignorance In which he suggested the universe is infinite ( the universe does not have a center, the pattern of stars would look the same at all locations. all motion is relative, & that the Earth might not be stationary Homogeneity & Relativity

27 Earth moves from Center Stage... The suggestion by Nicholas de Cusa (c.1430) that the Earth might not be stationary, was supported by Leonardo da Vinci (c1490), who amongst many (!) other things also suggested the Earth moves (rather than the Sun). However it was not until 1543 when Nicholas Copernicus publishes his Revolutionibus Orbium Coelestium (The Revolution of the Celestial Spheres) that this idea was put of a more rigorous footing.

28 Heliocentric Cosmology Copernicus suggested the planets rotate (on circles) around a central Sun ….with "slower" planets being further from the Sun. Copernicus also acknowledged the Earth rotates on its axis Heliocentric Cosmology

29 A “Good” (Simpler) Model The heliocentric model of Copernicus obviously could be used to make predictions, that could be compared to observations. It was simpler than the model of Ptolemy that it replaced. However, it’s predictions were not any better than those of Ptolemy’s model unless (much smaller) epicycles were added

30 The Cosmological Principles The Copernican Cosmological Principle This is sometimes simply referred to as simply “The Cosmological Principle” One important aspect of Copernicus’ work - he took his heliocentric model, went further and made a model for the cosmos by saying, lets assume several things, then use observations to test whether this is a good model Cosmological principles are the assumptions which allow us to deduce the whole of nature on the basis of the observable to the unobservable. Not surprisingly, any study of cosmological principles must combine elements of astronomy, physics and philosophy.

31 The Copernican Cosmo Principle The Copernican Cosmological Principle is a logical extension of the the Copernican theory that the Earth is not the center of the universe. Thus the Earth is not "special", thus the "laws of nature" on (or around) Earth are not special. It is essentially a philosophical requirement/simplification necessary/assumed for all modern cosmologies: - our laws of physics are otherwise "irrelevant"

32 The CCP itself Note that the statement "has/will always be so" refers to the universe continuing to display the properties of homogeneity & isotropy. The CCP does not imply that any actual observable parameter (e.g. the density of matter in the universe) will remain constant with time. Indeed, the CCP allows the properties of the universe to evolve with time, but states that at any given time the universe will be both homogeneous and isotropic (in 3-D space). On a large scale, the universe is both homogeneous and isotropic (in 3-D space), and has/will always be so. The Copernican Cosmological Principle is that

33 The CCP again Another way of expressing the Copernican Cosmological Principle is that... observers will see identical properties & laws everywhere- homogeneity will NOT see any preferred direction - isotropy i.e. this was the suggestion that we do not occupy a special place in the universe

34 Homogeneity/Isotropy homogeneous - same properties everywhere isotropic - no special direction, uniform in all directions homogeneous but not isotropic isotropic but not homogeneous

35 The CCP - an analogy A (small) sentient being living in the center of a "perfect" loaf of bread…! There may be obvious structure on small scales (air bubbles etc), but on the large scale the loaf can be considered uniform and isotropic The laws of physics (e.g. which caused the dough to rise) are the same throughout the loaf. The loaf might still be rising - but (in this perfect loaf) this happens uniformly & following then same laws throughout the loaf

36 The CCP Evidence for & against The best support for the Copernican Cosmological Principle is the Cosmic Microwave Background (CMB), which is isotropic to 1 part in 10 5 The obvious observational evidence against the Copernican Cosmological Principle seems to be the structure seen in the universe on a variety of scales (stars, galaxies, clusters, super-clusters..the cosmic web) This is why the qualifier "On a large scale.." is required to be added to the principle. The question them becomes a question of scale (now large is "large" ?), and whether the observed structures on large scales are indeed representative of the universe on these scales (or are "perturbations" which "happen” to be visible to us).

37 Recap We are up to 1400 We briefly mentioned the ideas of Nicholas de Cusa Homogeneity & Relativity We discussed the Heliocentric system of Nicholas Copernicus We then discussed the Copernican Cosmological Principle On a large scale, the universe is both homogeneous and isotropic (in 3-D space)

38 What have we learned? How can we distinguish science from non-science? It’s not always easy, but science generally exhibits at least three hallmarks. (1) Modern science seeks explanations for observed phenomena that rely solely on natural causes. (2) Science progresses through the creation and testing of models of nature that explain the observations as simply as possible (Ockhams Razor) (3) A scientific model must make testable predictions about natural phenomena that would force us to revise or abandon the model if the predictions do not agree with observations.

39 What have we learned? What is a theory in science? A model that explains a wide variety of observations in terms of just a few general principles, which has survived numerous tests to verify its predictions and explanations. How were astronomy and astrology related in the past, and are they still related today? Astronomy and astrology both grew out of ancient observations of the sky. Astronomy grew into a modern science. Astrology has never passed scientific tests and does not qualify as science (ALTHOUGH EARLY ON IT HELPED US DEVELOP ASTRONOMY).

40 What else have we learned? Copernicus created a Sun-centered model of the solar system designed to replace the Ptolemaic model, but was more fundamentally correct yet it was no more accurate because he still used perfect circles.

41 Tycho Brahe (c.1570)) Also famous for having lost his nose in a swordfight Tycho Brahe (c.1570) did accept that the (other) Planets move around the Sun but did not accept that the Earth & Stars move around the Sun Why ? … Falling bodies fall towards the Earth... The lack of Stellar Parallax

42 What was his problem ?) Falling Bodies fall towards the Earth Indeed if you throw something vertically upwards, it falls vertically downwards (to the same spot) Tycho Brahe reasoned this surely meant the Earth was the center of the universe Tycho Brahe was unable to detect (by naked-eye) Stellar Parallax and reasoned that in a Copernican system this would require the Stars to be so far away they would have to be "unreasonably" large/bright.

43 Tycho Brahe the observer Tycho Brahe was primarily an observer a Supernova position did not change (so it was not a comet or meteor), - lack of Parallax must be in one of the “outer spheres” -therefore the outer sphere of stars does change! a Comet position did not change significantly throughout the night. - lack of Parallax, must lay beyond the orbit of the Venus Observed positions of Mars twice-daily which implied its orbit intersects that of the Sun. - apparent crossing/smashing of the “crystalline spheres” “there are no solid spheres "holding" the celestial bodies” - observations that strengthened the rejection of the cosmology of Aristotle & Ptolemy

44 Tycho Brahe - his contribution So, even though Tycho Brahe never found the “right” model… … his observations did play a major role in the final rejection of the notions of Aristotle/Ptolemy that the celestial bodies are carried by crystalline spheres, with everything beyond the Moon eternal & unchanging. In addition Tycho Brahe also actually published his data ! In particular his twice-daily measurements of the position of Mars provided Johannes Kepler with a crucial database a few years later.

45 Johannes Kepler (1571-1630)  Greatest theorist of his day  a mystic  there were no heavenly spheres  forces made the planets move  Modifications to Copernicus idea showed heliocentric models were correct

46 Kepler’s Laws - #1 1 Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

47 Properties of an Ellipse a + b = constant The amount of "flattening" of the ellipse is termed the eccentricity All ellipses have eccentricities lying between zero and one. e=ratio; dist. between foci compared to major axis

48 Kepler’s Laws - #1 1 Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus.

49 Kepler's Laws: Kepler's Second Law: Line joining planet and the Sun sweeps out equal areas in equal times Kepler's First Law: 1 Each planet’s orbit around the Sun is an ellipse, with the Sun at one focus. Kepler's Third Law: The squares of the periods of the planets are proportional to the cubes of their semi-major axes:

50 Kepler’s Laws 2 A convenient unit of measurement for periods is in Earth years, and a convenient unit of measurement for distances is the average separation of the Earth from the Sun, which is termed an astronomical unit and is abbreviated as AU. If these units are used in Kepler's 3rd Law, the denominators in the preceding equation are numerically equal to unity and it may be written in the simple form This equation may then be solved for the period P of the planet, given the length of the semi-major axis axis, or for length of the semi-major axis, given the period of the planet Kepler’s 3rd Law: version 2

51 Kepler’s Laws 3 As an example of using Kepler's 3rd Law: 3 let's calculate the "radius" of the orbit of Mars (that is, the length of the semi-major axis of the orbit) from the orbital period. The time for Mars to orbit the Sun is observed to be 1.88 Earth years. Thus, by Kepler's 3rd Law the length of the semi-major axis for the Martian orbit is Second example, let us calculate the orbital period for Pluto, given that its observed average separation from the Sun is 39.44 astronomical units. From Kepler's 3rd Law

Categories: 1

0 Replies to “Orbitz And Keplers Laws Tutorial Mastering Astronomy Assignment”

Leave a comment

L'indirizzo email non verrà pubblicato. I campi obbligatori sono contrassegnati *